The Republic came before the Farragut; Kirk was an ensign aboard the former and a lieutenant aboard the latter. We know the Farragut lost its captain 11 years before "Obsession," or around 2256-7. Kirk was in command of the Enterprise by 2265. That's eight years to get from lieutenant on the Farragut to captain on the Enterprise, which doesn't seem too unreasonable considering that Kirk was supposed to be the youngest captain in Starfleet history (according to The Making of Star Trek, though never stated in canon). And it's consistent with other characters' Starfleet records; Will Riker was offered his first command only seven years out of the Academy.
To get from lieutenant on the Farragut to captain of the Enterprise in eight years, if we accept that Kirk was the youngest captain in Starfleet history, does seem doable. But is it doable to insert another command of another ship in there? That's what I was discussing, whether he had time to have served a tour as captain of ANOTHER ship before getting the Enterprise.